146 VII SPACES OF CONTINUOUS FUNCTIONS

(b) Sufficiency. Suppose / is the uniform limit of a sequence QQ of
step-functions. For each £>0 there is an n such that ||/-/ill <e/3;'
now for each x e I, there is an interval ]c, d[ containing x and such that
\\fn(s)~-fn(t)\\ <£/3 if both s and t are in ]c9 x[ or both in ]x,d[; hence
under the same assumption we have \\f(s) -/(OH < ^ ancl this proves the
existence of one-sided limits of/at x, since F is complete (3,14.6),

Another way of formulating (7.6.1) is to say that the set of regulated
functions is closed in ^F(E), and that the set of step-functions is dense in
the set of regulated functions.

(7.6.2) Any continuous mapping of an interval I c R into a JSanach space
is regulated; so is any monotone mapping of I into R,

This follows from the definition, taking into account (3.16.5) and (4.2,1),

PROBLEMS

1. Let/be a regulated mapping of an interval I c R into a Banach space F, Show that
for each compact subset H of I, /(H) is relatively compact in F; give an example
showing that /(H) need not be closed in F.

2. The function/W «= x sin(l/*) (/(O) = 0) is continuous, hence regulated In J [0, t],
and the function g(x) = sgn x (g(x) «= lif x > 0,0(0) " 0, #(*) -1 if .v • 0) is regu-
lated in R, but the composed function g o/is not regulated in I.

3. Let I = [a, b] be a compact interval in R. A function of bounded variation in I is a mapping
/of I into a Banach space F, having the following property; there is a number V "r. 0
such that, for any strictly increasing finite sequence (//)0 «i «„ of points of I»the Inequality

E ll/(f/+i)-/(/,)KV holds.

f = 0

(a) Show that /(I) is relatively compact in F (prove that /(I) is precompact, by an
indirect proof).

(b) Show that/is a regulated function in I (use (a) and (3.16*4)).

(c) The function g defined in [0,1] as equal to x2 sin(l/jea) for x 9* 0 and to 0 for

* = 0 is not of bounded variation, although it has a derivative at each point of I.e points a* IK xt, y(xt) and z(.V|). As each cj is in some